A theoretical investigation of rates of diffusion-limited polymerization between rigid rodlike molecules in dilute solutions is presented, incorporating both translational and rotational diffusion of the molecules. Simplified criteria for reaction are proposed according to which molecules must approach within a specified reaction radius, and have a relative orientation angle in a specified range to react. The translational diffusion of the molecules is assumed to be isotropic. The results of the exact analytical solution, finite element computations, as well as approximate solutions for low rotational diffusivity, are presented. The match between the solutions is satisfactory in the appropriate limits, although the exact solution is found to be computationally intensive for low rotational diffusivities and rapid intrinsic kinetics. The reaction rate is found to decrease to a limiting value with decreasing rotational diffusivity, and the magnitude of the decrease in rate increases with intrinsic rate of reaction. The limiting value of rate is entirely due to the translational diffusive flux of the correctly oriented molecules from the far field, and an asymptotic expression for the rate in this limit is given. Comparison with reported experimental data is qualitative, since the data is only available for the semidilute regime for such systems, where the anisotropic translational diffusion is expected to play a prominent role.